A leafwise positive closed two-form calibrates a taut foliation

Suppose $F$ is not taut. By [F1] there is a dead-end component $N$ whose closure $\bar{N}$ is a compact oriented three-manifold with boundary a finite union of compact leaves $L_i$ along which the co-orientation points inwards.

Positivity on $TF$ makes $\theta$ a rank-two form everywhere. Its kernel $K$ is a smooth line bundle transverse to $TF$: if a nonzero tangent vector of a leaf lay in $K$, its contraction with the positive area form $\theta {\mathrm{\mid }}_{TF}$ would not vanish. Choose a smooth metric $h$ on $TF$ by F4 and write $\theta {\mathrm{\mid }}_{TF}=a{\mu }_h$ with smooth $a>0$. In dimension two the metric $ah$ has area form $a{\mu }_h$. Give $K$ any smooth metric and declare $TF\perp K$, obtaining a smooth ambient metric. Since $\theta$ annihilates $K$ and equals the unit area form on $TF$, its value on any unit simple two-vector has absolute value at most one, by the determinant bound for orthogonal projection onto the two-plane $TF$. Equality holds on the oriented unit leaf tangent bivector. This explicitly proves the comass-one calibration assertion; an arbitrary previously chosen transverse line would not have eliminated mixed components of $\theta$.

Stokes gives ${\sum }_i\pm {\int }_{L_i}\theta ={\int }_{\bar{N}}d\theta =0$, because $\theta$ is closed. Each boundary leaf carries the orientation induced from the co-orientation and the orientation of $M$ by [F2]; since the co-orientation points inwards on every boundary component, all signs in the sum coincide, so all integrals ${\int }_{L_i}\theta$ have the same sign. Each integral is nonzero because $\theta {\mathrm{\mid }}_{L_i}$ is a positive area form on the compact leaf $L_i$ by the positivity hypothesis, so by [F3] every integral is strictly positive for the induced orientation. Hence the sum cannot vanish, a contradiction; therefore no dead-end component exists and $F$ is taut.

Let $D$ be a compact smooth domain in a leaf and vary its immersion by a compactly supported ambient normal field that vanishes near $\mathrm{\partial }D$. Stokes applied to the homotopy cylinder gives ${\int }_D{F}_t^{\ast }\theta ={\int }_D{F}_0^{\ast }\theta$, because $d\theta =0$ and the cylinder's side is fixed. The comass bound from step 1.2 gives $\mathrm{Area}(F_t{\mathrm{\mid }}_D)\ge {\int }_D{F}_t^{\ast }\theta =\mathrm{Area}(F_0{\mathrm{\mid }}_D)$ for both signs of small $t$. Hence its first derivative is zero. By F4, ${\int }_D⟨V,H⟩=0$ for every such normal variation. Locally extend $V=\eta H$, with any nonnegative bump $\eta$ supported in a small embedded leaf chart, to a compactly supported ambient normal field; F4 supplies the flow realizing it. Then $\int \eta \mathrm{\mid }H{\mathrm{\mid }}^2=0$, so continuity and arbitrary bumps imply $H=0$ everywhere. This proves minimality for every leaf, including noncompact leaves, without importing a calibration-to-minimality theorem.

Combining the preceding steps, a closed $2$-form positive on the leaves forces tautness and calibrates the foliation, with every leaf minimal; the argument uses one Stokes computation and finitely many local metric choices, hence only the standing countable choice from [F5].